Angle Modulation: Frequency Modulation (FM), Phase Modulation (PM), and Carson Rule

Unit 3: Amplitude modulation and synchronous detection

Learning objective: Analyze conventional AM, suppressed-carrier DSB and SSB by tracking the spectrum and detector assumptions.

Physical perspective

Analog amplitude modulation illustrates the cost of carrying information over a bandpass channel. A carrier makes envelope detection simpler but consumes transmitted power that carries no independent message samples. Suppressing the carrier improves power allocation yet requires phase-coherent downconversion. A single-sideband signal saves occupied bandwidth at the cost of more elaborate generation and receiver tuning.

Extended engineering case: how to reason from requirements

Amplitude modulation is easiest to understand through conservation of spectral content. Multiplying a baseband message by a carrier produces upper and lower translated copies. Including an unmodulated carrier changes the power distribution and creates an envelope that a simple circuit can track. These alternatives involve different receiver assumptions. DSB-SC and SSB save carrier power but a coherent receiver must know the oscillator reference. Conventional AM is tolerant of phase uncertainty at the envelope detector but transmits a carrier even when the message is silent. In SSB the result of an oscillator frequency error is an audible pitch shift rather than merely a constant gain loss.

A field-service engineer might use an amplitude-modulated test beacon to probe a narrowband receiver. Before attributing a distorted recovered waveform to the air path, verify that the source modulation index did not exceed unity, the envelope detector RC time constant was appropriate and the measurement input did not overload. A detector with a capacitor too small follows high-frequency ripple, whereas a time constant too large cannot follow a rapidly decreasing envelope and causes diagonal clipping. The correct compromise depends on both carrier and maximum message frequency. A product detector has a different failure mode: carrier-phase error reduces the in-phase branch even if the envelope itself is perfectly formed.

The simulator in this unit deliberately isolates those failures. Its AM sweep should show an envelope crossing zero when the normalized single-tone modulation index exceeds one. Do not call a clipped trace a measurement of over-the-air broadcast quality. Add finite filter settling and a carrier-offset thought experiment to the lab record. When comparing occupied bandwidth, identify whether the message bandwidth refers to a one-sided positive-frequency interval or a two-sided baseband spectrum.

Connections and study order

The next four lessons move from the simplest tractable model to a broader engineering treatment of this unit. Solve their worked cases separately first, then assemble the assumptions in a single coherent link design. For every step, distinguish data representing the actual message from a signal generated by the transceiver and from noise introduced by the environment or receiver. Use the model's limiting cases to decide whether parameters can be safely neglected. Finally, specify how any simulation could be validated through an independent analytic calculation and, if qualified laboratory access exists, through a supervised physical measurement.

Unit lessons

• [3.1 Double-sideband suppressed-carrier modulation](../lessons/U03-L01-double-sideband-suppressed-carrier-modulation.html) — model s(t)=m(t)cos(2πfct) shifts M(f) to ±fc, each scaled by 1/2
• [3.2 Conventional AM depth and envelope detection](../lessons/U03-L02-conventional-am-depth-and-envelope-detection.html) — model s(t)=Ac[1+μ m_norm(t)]cos(2πfct), |m_norm|≤1
• [3.3 Single-sideband and analytic signals](../lessons/U03-L03-single-sideband-and-analytic-signals.html) — model analytic m_a(t)=m(t)+j H{m(t)}; SSB=Re{m_a(t)e^(j2πfct)}
• [3.4 Carrier recovery and phase-sensitive detector error](../lessons/U03-L04-carrier-recovery-and-phase-sensitive-detector-error.html) — model low-pass{2m(t)cos(ωct)cos(ωct+φ)}=m(t)cosφ

Unit assessment and project

• [Self-assessment](../assessments/unit-03.html)
• [Integrated project](../projects/unit-03.html)
• [Interactive model](../simulations/model-03.html)

Lesson 3.1 Double-sideband suppressed-carrier modulation

Unit: 3 — Amplitude modulation and synchronous detection Core model: s(t)=m(t)cos(2πfct) shifts M(f) to ±fc, each scaled by 1/2 Suggested study: derive the model, solve its worked case, compare one limiting case, then attempt the open questions.

What is being described?

Multiplication in time is convolution in frequency, so a baseband spectrum creates mirrored copies around positive and negative carrier centers. A coherent receiver multiplies by the correct local oscillator and filters away the doubled carrier terms. For this convention a gain factor of two may be required to recover the original message amplitude, depending on transmitter and receiver scaling. If transmitter and receiver oscillators differ, the recovered message undergoes a residual low-frequency rotation or beating.

Derivation and interpretation

Begin by writing the input, output and channel reference planes for double-sideband suppressed-carrier modulation. The governing relationship in this lesson is s(t)=m(t)cos(2πfct) shifts M(f) to ±fc, each scaled by 1/2. Read each symbol according to its definition in the accompanying discussion, not as a context-free calculator instruction. Where the formula uses decibels, convert back to linear power before adding contributions. Where it describes a spectrum, identify whether positive and negative frequencies, peak or RMS amplitudes and window normalization are being used. If a formula describes probability, specify the random variable and statistical model. These conditions are substantive mathematical requirements: using the right arithmetic on the wrong reference conditions does not produce a correct engineering result.

Worked engineering case

For m(t)=cos(2π·1 kHz·t) and fc=100 kHz, the transmitted real signal contains components at 99 and 101 kHz. A 100 kHz product detector creates the original 1 kHz term plus high-frequency terms that the low-pass filter rejects.

Failure mode and model boundary

A passive low-pass filter cannot fix a phase error of exactly 90 degrees in a one-branch synchronous detector; the useful component can vanish.

Investigation and design transfer

Generate a DSB-SC tone at two message frequencies. Explain the measured sideband locations and why a spectrum display should not show a large carrier line.

Questions and solution guidance

1. Derivation: Starting from the stated governing relation, define the variables and derive or justify the result under its intended assumptions. Guidance: show every conversion, distinguish the signal and noise reference planes, and explain why each approximation holds.

2. Check the worked example: Recompute the numerical case, then change one parameter by a factor of two. Guidance: perform the change in linear units when powers or probabilities are involved; translate to decibels only at the end, if appropriate.

3. Challenge the assumption: A passive low-pass filter cannot fix a phase error of exactly 90 degrees in a one-branch synchronous detector; the useful component can vanish. Guidance: identify the precise assumption being challenged and propose an observable counter-test rather than treating the limitation as a generic warning.

4. Engineering design: Generate a DSB-SC tone at two message frequencies. Explain the measured sideband locations and why a spectrum display should not show a large carrier line. Guidance: specify achievable measurements, independent variables, expected plots and a criterion for accepting or rejecting the design.

Connection to the rest of the chain

A communication receiver does not experience this mechanism in isolation. Source statistics influence coding; waveform choice influences occupied bandwidth; filtering changes the noise entering the decision statistic; and synchronization transforms signal geometry into errors. When using the result from double-sideband suppressed-carrier modulation inside an end-to-end budget, name its upstream inputs and downstream consequences. Avoid adding an improvement twice: reducing noise bandwidth and improving SNR through a filter may describe the same physical change. Conversely, an unmodeled impairment can create a gap between a correct local calculation and actual delivered payload. A full design review therefore asks for both local correctness and end-to-end error or outage evidence.

Lesson 3.2 Conventional AM depth and envelope detection

Unit: 3 — Amplitude modulation and synchronous detection Core model: s(t)=Ac[1+μ m_norm(t)]cos(2πfct), |m_norm|≤1 Suggested study: derive the model, solve its worked case, compare one limiting case, then attempt the open questions.

What is being described?

When the normalized message peaks between minus one and plus one, a modulation index no greater than one keeps the envelope nonnegative. An envelope detector can then track the intended amplitude without knowing carrier phase, assuming its RC response is fast relative to message changes but slow relative to carrier cycles. Overmodulation reverses envelope sign and produces severe nonlinear distortion for a diode detector. For single-tone AM the sideband powers sum to a fraction μ²/2 of carrier power; this is a statement about the conventional ideal single-tone waveform.

Derivation and interpretation

Begin by writing the input, output and channel reference planes for conventional am depth and envelope detection. The governing relationship in this lesson is s(t)=Ac[1+μ m_norm(t)]cos(2πfct), |m_norm|≤1. Read each symbol according to its definition in the accompanying discussion, not as a context-free calculator instruction. Where the formula uses decibels, convert back to linear power before adding contributions. Where it describes a spectrum, identify whether positive and negative frequencies, peak or RMS amplitudes and window normalization are being used. If a formula describes probability, specify the random variable and statistical model. These conditions are substantive mathematical requirements: using the right arithmetic on the wrong reference conditions does not produce a correct engineering result.

Worked engineering case

At μ=0.8, total transmitted power relative to carrier-only power is 1+0.8²/2=1.32; the ideal sideband fraction of total power is 0.32/1.32≈24.24%.

Failure mode and model boundary

Power efficiency depends on the modulation waveform and definition. Avoid applying the single-tone power fraction unchanged to speech whose rms message is not 1/√2.

Investigation and design transfer

Examine μ=0.5, 1.0 and 1.2. Identify which simulated envelope crosses zero and explain why the last case cannot be recovered faithfully by a simple envelope detector.

Questions and solution guidance

1. Derivation: Starting from the stated governing relation, define the variables and derive or justify the result under its intended assumptions. Guidance: show every conversion, distinguish the signal and noise reference planes, and explain why each approximation holds.

2. Check the worked example: Recompute the numerical case, then change one parameter by a factor of two. Guidance: perform the change in linear units when powers or probabilities are involved; translate to decibels only at the end, if appropriate.

3. Challenge the assumption: Power efficiency depends on the modulation waveform and definition. Avoid applying the single-tone power fraction unchanged to speech whose rms message is not 1/√2. Guidance: identify the precise assumption being challenged and propose an observable counter-test rather than treating the limitation as a generic warning.

4. Engineering design: Examine μ=0.5, 1.0 and 1.2. Identify which simulated envelope crosses zero and explain why the last case cannot be recovered faithfully by a simple envelope detector. Guidance: specify achievable measurements, independent variables, expected plots and a criterion for accepting or rejecting the design.

Connection to the rest of the chain

A communication receiver does not experience this mechanism in isolation. Source statistics influence coding; waveform choice influences occupied bandwidth; filtering changes the noise entering the decision statistic; and synchronization transforms signal geometry into errors. When using the result from conventional am depth and envelope detection inside an end-to-end budget, name its upstream inputs and downstream consequences. Avoid adding an improvement twice: reducing noise bandwidth and improving SNR through a filter may describe the same physical change. Conversely, an unmodeled impairment can create a gap between a correct local calculation and actual delivered payload. A full design review therefore asks for both local correctness and end-to-end error or outage evidence.

Lesson 3.3 Single-sideband and analytic signals

Unit: 3 — Amplitude modulation and synchronous detection Core model: analytic m_a(t)=m(t)+j H{m(t)}; SSB=Re{m_a(t)e^(j2πfct)} Suggested study: derive the model, solve its worked case, compare one limiting case, then attempt the open questions.

What is being described?

A Hilbert-transform pair allows one sideband of a real modulation to cancel while the other is retained. In the frequency domain this is the analytic-signal construction: negative-frequency content of the complex baseband is suppressed and positive-frequency components are doubled under a standard convention. The final physical waveform remains real. A practical SSB receiver uses a beat-frequency oscillator and narrowband filtering; small frequency mismatch shifts every recovered audio component and may make a voice unnatural.

Derivation and interpretation

Begin by writing the input, output and channel reference planes for single-sideband and analytic signals. The governing relationship in this lesson is analytic m_a(t)=m(t)+j H{m(t)}; SSB=Re{m_a(t)e^(j2πfct)}. Read each symbol according to its definition in the accompanying discussion, not as a context-free calculator instruction. Where the formula uses decibels, convert back to linear power before adding contributions. Where it describes a spectrum, identify whether positive and negative frequencies, peak or RMS amplitudes and window normalization are being used. If a formula describes probability, specify the random variable and statistical model. These conditions are substantive mathematical requirements: using the right arithmetic on the wrong reference conditions does not produce a correct engineering result.

Worked engineering case

A message occupying 300–3000 Hz needs approximately 2.7 kHz of idealized SSB information bandwidth, rather than two mirrored 2.7 kHz sidebands for conventional double-sideband transmission. Allow extra room for transmitter and filter skirts in practical allocations.

Failure mode and model boundary

A sharp ideal Hilbert transform has nontrivial delay and edge-response implications; real finite filters never have perfect suppression over all frequencies.

Investigation and design transfer

Compare upper- and lower-sideband spectral occupancy for a two-tone message. Provide the expected four candidate sideband locations before cancellation.

Questions and solution guidance

1. Derivation: Starting from the stated governing relation, define the variables and derive or justify the result under its intended assumptions. Guidance: show every conversion, distinguish the signal and noise reference planes, and explain why each approximation holds.

2. Check the worked example: Recompute the numerical case, then change one parameter by a factor of two. Guidance: perform the change in linear units when powers or probabilities are involved; translate to decibels only at the end, if appropriate.

3. Challenge the assumption: A sharp ideal Hilbert transform has nontrivial delay and edge-response implications; real finite filters never have perfect suppression over all frequencies. Guidance: identify the precise assumption being challenged and propose an observable counter-test rather than treating the limitation as a generic warning.

4. Engineering design: Compare upper- and lower-sideband spectral occupancy for a two-tone message. Provide the expected four candidate sideband locations before cancellation. Guidance: specify achievable measurements, independent variables, expected plots and a criterion for accepting or rejecting the design.

Connection to the rest of the chain

A communication receiver does not experience this mechanism in isolation. Source statistics influence coding; waveform choice influences occupied bandwidth; filtering changes the noise entering the decision statistic; and synchronization transforms signal geometry into errors. When using the result from single-sideband and analytic signals inside an end-to-end budget, name its upstream inputs and downstream consequences. Avoid adding an improvement twice: reducing noise bandwidth and improving SNR through a filter may describe the same physical change. Conversely, an unmodeled impairment can create a gap between a correct local calculation and actual delivered payload. A full design review therefore asks for both local correctness and end-to-end error or outage evidence.

Lesson 3.4 Carrier recovery and phase-sensitive detector error

Unit: 3 — Amplitude modulation and synchronous detection Core model: low-pass{2m(t)cos(ωct)cos(ωct+φ)}=m(t)cosφ Suggested study: derive the model, solve its worked case, compare one limiting case, then attempt the open questions.

What is being described?

Coherent demodulation works by aligning a receiver-generated carrier with the transmitted signal. A constant phase difference reduces the recovered amplitude for a scalar DSB detector; time-varying phase introduces modulation artifacts and spectral spreading. An I/Q receiver retains two components and can estimate phase without depending on a single cosine branch. A phase-locked loop uses error feedback to track oscillator frequency and phase over a limited range and dynamic response.

Derivation and interpretation

Begin by writing the input, output and channel reference planes for carrier recovery and phase-sensitive detector error. The governing relationship in this lesson is low-pass{2m(t)cos(ωct)cos(ωct+φ)}=m(t)cosφ. Read each symbol according to its definition in the accompanying discussion, not as a context-free calculator instruction. Where the formula uses decibels, convert back to linear power before adding contributions. Where it describes a spectrum, identify whether positive and negative frequencies, peak or RMS amplitudes and window normalization are being used. If a formula describes probability, specify the random variable and statistical model. These conditions are substantive mathematical requirements: using the right arithmetic on the wrong reference conditions does not produce a correct engineering result.

Worked engineering case

At a fixed 60 degree phase error, the recovered amplitude of an ideal coherent DSB branch is cos60°=0.5 of its phase-aligned value. At 90 degrees it is ideally zero. A time-varying 1 Hz frequency mismatch causes the relative phase to drift by one cycle each second.

Failure mode and model boundary

A PLL is not an arbitrary repair tool: acquisition range, noise bandwidth, cycle slips and pull-in time matter, especially at weak SNR.

Investigation and design transfer

Design a carrier-recovery test with a swept phase offset, then add a slow frequency offset. Explain differences between static phase loss and a continually rotating constellation.

Questions and solution guidance

1. Derivation: Starting from the stated governing relation, define the variables and derive or justify the result under its intended assumptions. Guidance: show every conversion, distinguish the signal and noise reference planes, and explain why each approximation holds.

2. Check the worked example: Recompute the numerical case, then change one parameter by a factor of two. Guidance: perform the change in linear units when powers or probabilities are involved; translate to decibels only at the end, if appropriate.

3. Challenge the assumption: A PLL is not an arbitrary repair tool: acquisition range, noise bandwidth, cycle slips and pull-in time matter, especially at weak SNR. Guidance: identify the precise assumption being challenged and propose an observable counter-test rather than treating the limitation as a generic warning.

4. Engineering design: Design a carrier-recovery test with a swept phase offset, then add a slow frequency offset. Explain differences between static phase loss and a continually rotating constellation. Guidance: specify achievable measurements, independent variables, expected plots and a criterion for accepting or rejecting the design.

Connection to the rest of the chain

A communication receiver does not experience this mechanism in isolation. Source statistics influence coding; waveform choice influences occupied bandwidth; filtering changes the noise entering the decision statistic; and synchronization transforms signal geometry into errors. When using the result from carrier recovery and phase-sensitive detector error inside an end-to-end budget, name its upstream inputs and downstream consequences. Avoid adding an improvement twice: reducing noise bandwidth and improving SNR through a filter may describe the same physical change. Conversely, an unmodeled impairment can create a gap between a correct local calculation and actual delivered payload. A full design review therefore asks for both local correctness and end-to-end error or outage evidence.

Unit 3 project — Amplitude modulation and synchronous detection: an auditable synthetic design study

Problem statement. Analyze conventional AM, suppressed-carrier DSB and SSB by tracking the spectrum and detector assumptions. The deliverable is a self-contained engineering analysis, not a claim that a device was measured or certified.

Scenario

Amplitude modulation is easiest to understand through conservation of spectral content. Multiplying a baseband message by a carrier produces upper and lower translated copies. Including an unmodulated carrier changes the power distribution and creates an envelope that a simple circuit can track. These alternatives involve different receiver assumptions. DSB-SC and SSB save carrier power but a coherent receiver must know the oscillator reference. Conventional AM is tolerant of phase uncertainty at the envelope detector but transmits a carrier even when the message is silent. In SSB the result of an oscillator frequency error is an audible pitch shift rather than merely a constant gain loss.

Required work products

1. Requirements document. State the message type, sample or symbol rate, occupied-bandwidth convention, reference impedances or powers and a numerical reliability criterion. Differentiate functional requirements (what users need) from implementation choices (how blocks operate).

2. Equation notebook. For each of the unit's four governing relationships, derive the formula assumptions, provide an annotated worked calculation and evaluate a limiting case. Keep all intermediate calculations in linear units until a logarithmic representation is actually necessary.

3. Simulation experiment. Open the associated synthetic model. Take screenshots or download CSV for three input settings: nominal, stressed, and nearly ideal. Record why each setting was chosen. Compare at least two numerical points with a direct formula calculation independent of the browser code.

4. Failure analysis. Write one paragraph each on systematic bias, random variation and model mismatch. State the measurement instrument or independent experiment needed to distinguish the alternatives in a supervised hardware implementation.

5. Decision memo. Recommend a configuration for the stated application. Include a tradeoff table of bandwidth, power, complexity, delay, expected errors and uncertainty. Explicitly identify the two most consequential unverified assumptions.

Suggested numerical investigation

• Experiment 1, Double-sideband suppressed-carrier modulation: Generate a DSB-SC tone at two message frequencies. Explain the measured sideband locations and why a spectrum display should not show a large carrier line. Translate the experiment into three adjustable parameters; record predicted trend and one case that would challenge it.
• Experiment 2, Conventional AM depth and envelope detection: Examine μ=0.5, 1.0 and 1.2. Identify which simulated envelope crosses zero and explain why the last case cannot be recovered faithfully by a simple envelope detector. Translate the experiment into three adjustable parameters; record predicted trend and one case that would challenge it.
• Experiment 3, Single-sideband and analytic signals: Compare upper- and lower-sideband spectral occupancy for a two-tone message. Provide the expected four candidate sideband locations before cancellation. Translate the experiment into three adjustable parameters; record predicted trend and one case that would challenge it.
• Experiment 4, Carrier recovery and phase-sensitive detector error: Design a carrier-recovery test with a swept phase offset, then add a slow frequency offset. Explain differences between static phase loss and a continually rotating constellation. Translate the experiment into three adjustable parameters; record predicted trend and one case that would challenge it.

Evaluation rubric (100 points)

• Model derivations and units: 25 points. A formula with unexplained symbols does not earn full marks.
• Independent arithmetic and numerical consistency: 20 points. Reproduce at least one graph point without reusing the graph's implementation.
• Simulation experiment and transparent provenance: 20 points. Identify every result as synthetic.
• Failure analysis and sensitivity: 20 points. Include meaningful nonideal tests, not only cosmetic parameter changes.
• Clear engineering recommendation: 15 points. Tie the recommendation to the defined objective and state where further evidence is required.

Safety: do not build or operate radio transmitters, powered antennas or high-voltage test apparatus as part of this offline project.

Unit 3 assessment — Amplitude modulation and synchronous detection

Instructions: show the reference planes, numerical steps and units. These are explanatory solution outlines, not a substitute for deriving the equations. Allocate 100 points: 20 each for Questions 1–4 and 20 for the integration problem.

Question 1: Double-sideband suppressed-carrier modulation (20 points)

Problem. For m(t)=cos(2π·1 kHz·t) and fc=100 kHz, the transmitted real signal contains components at 99 and 101 kHz. A 100 kHz product detector creates the original 1 kHz term plus high-frequency terms that the low-pass filter rejects. Reconstruct the calculation or argument and then choose a plausible change of one operating variable. Explain the direction and approximate magnitude of the result without assuming the model applies outside its stated range. Give one separate observation that would invalidate the ideal prediction.

Solution outline. Begin with s(t)=m(t)cos(2πfct) shifts M(f) to ±fc, each scaled by 1/2. A passive low-pass filter cannot fix a phase error of exactly 90 degrees in a one-branch synchronous detector; the useful component can vanish. Full credit requires the value or directional consequence, defined units, and explicit treatment of the limiting assumptions. A correct numerical answer without a meaningful reference plane receives at most half credit; a diagram without a justified computation also receives partial credit only.

Question 2: Conventional AM depth and envelope detection (20 points)

Problem. At μ=0.8, total transmitted power relative to carrier-only power is 1+0.8²/2=1.32; the ideal sideband fraction of total power is 0.32/1.32≈24.24%. Reconstruct the calculation or argument and then choose a plausible change of one operating variable. Explain the direction and approximate magnitude of the result without assuming the model applies outside its stated range. Give one separate observation that would invalidate the ideal prediction.

Solution outline. Begin with s(t)=Ac[1+μ m_norm(t)]cos(2πfct), |m_norm|≤1. Power efficiency depends on the modulation waveform and definition. Avoid applying the single-tone power fraction unchanged to speech whose rms message is not 1/√2. Full credit requires the value or directional consequence, defined units, and explicit treatment of the limiting assumptions. A correct numerical answer without a meaningful reference plane receives at most half credit; a diagram without a justified computation also receives partial credit only.

Question 3: Single-sideband and analytic signals (20 points)

Problem. A message occupying 300–3000 Hz needs approximately 2.7 kHz of idealized SSB information bandwidth, rather than two mirrored 2.7 kHz sidebands for conventional double-sideband transmission. Allow extra room for transmitter and filter skirts in practical allocations. Reconstruct the calculation or argument and then choose a plausible change of one operating variable. Explain the direction and approximate magnitude of the result without assuming the model applies outside its stated range. Give one separate observation that would invalidate the ideal prediction.

Solution outline. Begin with analytic m_a(t)=m(t)+j H{m(t)}; SSB=Re{m_a(t)e^(j2πfct)}. A sharp ideal Hilbert transform has nontrivial delay and edge-response implications; real finite filters never have perfect suppression over all frequencies. Full credit requires the value or directional consequence, defined units, and explicit treatment of the limiting assumptions. A correct numerical answer without a meaningful reference plane receives at most half credit; a diagram without a justified computation also receives partial credit only.

Question 4: Carrier recovery and phase-sensitive detector error (20 points)

Problem. At a fixed 60 degree phase error, the recovered amplitude of an ideal coherent DSB branch is cos60°=0.5 of its phase-aligned value. At 90 degrees it is ideally zero. A time-varying 1 Hz frequency mismatch causes the relative phase to drift by one cycle each second. Reconstruct the calculation or argument and then choose a plausible change of one operating variable. Explain the direction and approximate magnitude of the result without assuming the model applies outside its stated range. Give one separate observation that would invalidate the ideal prediction.

Solution outline. Begin with low-pass{2m(t)cos(ωct)cos(ωct+φ)}=m(t)cosφ. A PLL is not an arbitrary repair tool: acquisition range, noise bandwidth, cycle slips and pull-in time matter, especially at weak SNR. Full credit requires the value or directional consequence, defined units, and explicit treatment of the limiting assumptions. A correct numerical answer without a meaningful reference plane receives at most half credit; a diagram without a justified computation also receives partial credit only.

Question 5: integrate the unit into a receiver chain (20 points)

Problem. Propose a complete signal chain for the unit's principal application. Show where source content, oscillator/reference information, channel distortion, random noise and decision logic enter. For each of the four lesson mechanisms, label an observable that a student could calculate or simulate. Choose one validation check independent of the simulation itself.

Solution outline. A strong answer includes a correctly ordered block diagram, a table of input/output units, a resource budget (power, spectrum or sampling), and at least one numerical limiting case. A student's independent calculation must use the same model conventions as the interactive graph. Discuss which missing impairment could overturn the proposed design. The answer should explain why a locally accurate mechanism does not guarantee successful end-to-end information delivery.

Marking: 5 points for the architecture, 5 for model consistency, 5 for independent validation, 5 for credible limitations.